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Mathematics 6.

Table of contents
Divisibility
How to solve problems?
Rational numbers I.
Axial symmetry
Rational numbers II.
Fractions (revision)
Operations with fractions (revision)
Negative fractions
Multiplying fractions by fractions
The reciprocals of numbers
Division by a fraction
Properties of operations
Rational numbers
Mixed exercises
Proportion
Percentages
Probability, statistics
Supplementary material
Properties of operations
Example 1
Frank walks his dog for \latex{\frac{2}{3}} of an \latex{ hour } every morning and \latex{\frac{4}{5}} of an \latex{ hour } every evening. How many \latex{ hours } does he walk the dog in \latex{ 5 } \latex{ days }?
Solution
They walk \latex{\frac{2}{3}+\frac{4}{5}} \latex{ hours } a \latex{ day. }
In 5 \latex{ days }: \latex{5\times \left(\frac{2}{3}+\frac{4}{5} \right)=5\times \left(\frac{10}{15}+\frac{12}{15} \right)=\overset{1}{\cancel{5}}\times \frac{22}{\underset{3}{\cancel{15}} }=\frac{22}{3}=7\frac{1}{3}} . 
Calculating in another way: in \latex{ 5 } \latex{ days }, they walk \latex{5\times \frac{2}{3}} \latex{ hours } in the mornings and \latex{5\times \frac{4}{5}} \latex{ hours } in the evenings.
\latex{5\times \dfrac{2}{3}+\overset{1}{\cancel{5}}\times \dfrac{4}{\underset{1}{\cancel{5}} }=\dfrac{10}{3}+4=3\dfrac{1}{3}+4=7\dfrac{1}{3}.}
In \latex{ 5 } \latex{ days }, Frank spends  \latex{7\frac{1}{3}} \latex{ hours } walking his dog.
In the case of fractions, apply what you have learned about the order of operations.
  1. First, perform the operations between brackets.
  2. Perform the multiplications and divisions from left to right.
  3. Perform the additions and subtractions from left to right.
\latex{\left(\dfrac{3}{2}-\dfrac{5}{4} \right)\times \left(\dfrac{2}{3}-\dfrac{5}{6} \right)=\dfrac{6-5}{4} \times \dfrac{4-5}{6}=\dfrac{1}{4}\times \left(-\dfrac{1}{6} \right)=-\dfrac{1}{24}.}

\latex{\dfrac{1}{4}\times \dfrac{1}{3}\ -\dfrac{1}{6}\times \dfrac{5}{2}=\dfrac{1}{12}-\dfrac{5}{12}=-\dfrac{\overset{1}{\cancel{4}} }{\underset{3}{\cancel{12}} }=-\dfrac{1}{3}.}

\latex{\dfrac{7}{3}-\dfrac{4}{5}\div \dfrac{12}{25}=\dfrac{7}{3}-\dfrac{\overset{1}{\cancel{4}} }{\underset{1}{\cancel{5}} }\times \dfrac{\overset{5}{\cancel{25}} }{\underset{3}{\cancel{12}} }=\dfrac{7}{3}-\dfrac{5}{3}=\dfrac{2}{3}.}
The properties of operations are valid for fractions as well.
  1. The addends are interchangeable.
\latex{\begin{aligned}\begin{rcases}\textcolor{#2AB7EC}{\left(-\dfrac{1}{4} \right)}+\textcolor{#F56060}{\dfrac{2}{5}}=-\dfrac{5}{20}+\dfrac{8}{20}&=\dfrac{3}{20} \\ \textcolor{#f56060}{\dfrac{2}{5}} +\textcolor{#2AB7EC}{\left(-\dfrac{1}{4} \right)}=\dfrac{8}{20}-\dfrac{5}{20}&=\dfrac{3}{20} \end{rcases}\end{aligned} \textcolor{#2AB7EC}{-\dfrac{1}{4}}+\textcolor{#f56060}{\dfrac{2}{5}}=\textcolor{#f56060}{\dfrac{2}{5}}+\textcolor{#2AB7EC}{\left(-\dfrac{1}{4} \right)}}
\latex{a+b=b+a}
  1. The factors of any multiplication are interchangeable.
\latex{\begin{aligned}\begin{rcases}\textcolor{#2ab7ec}{\left(-\dfrac{1}{4} \right)}\times \textcolor{#f65060}{\dfrac{2}{5}}=-\dfrac{1}{\underset{2}{\cancel{4}} }\times \dfrac{\overset{1}{\cancel{2}} }{5}&=-\dfrac{1}{10} \\ \textcolor{#f65060}{\dfrac{2}{5}}\times \textcolor{#2ab7ec}{\left(-\dfrac{1}{4} \right)}=\dfrac{\overset{1}{\cancel{2}}}{5}\times \left(-\dfrac{1}{\underset{2}{\cancel{4}}} \right) &=-\dfrac{1}{10} \end{rcases}\end{aligned} \textcolor{#2ab7ec}{\left(-\dfrac{1}{4} \right)}\times \textcolor{#f65060}{\dfrac{2}{5}}=\textcolor{#f65060}{\dfrac{2}{5}}\times \textcolor{#2ab7ec}{\left(-\dfrac{1}{4} \right)}}
\latex{a\times b=b\times a}
  1. The addends can be grouped arbitrarily.
\latex{\begin{rcases}\begin{aligned} \textcolor{#2ab7ec}{\left(\dfrac{3}{5}+\dfrac{7}{10} \right)}-\dfrac{1}{2}=\left(\dfrac{6}{10}+\dfrac{7}{10} \right)-\dfrac{5}{10}=\dfrac{13}{10}-\dfrac{5}{10}=\dfrac{8}{10}&=\dfrac{4}{5} \\[10pt] \dfrac{3}{5}+\textcolor{#2ab7ec}{\left(\dfrac{7}{10}-\dfrac{1}{2} \right)}=\dfrac{6}{10}+\left(\dfrac{7}{10}-\dfrac{5}{10} \right)=\dfrac{6}{10}+\dfrac{2}{10}=\dfrac{8}{10}&=\dfrac{4}{5} \\[10pt] \dfrac{3}{5}+\dfrac{7}{10}-\dfrac{1}{2}=\dfrac{6}{10}+\dfrac{7}{10}-\dfrac{5}{10}=\dfrac{8}{10}&=\dfrac{4}{5} \end{aligned}\end{rcases}}
that is
\latex{\textcolor{#2ab7ec}{\left(\frac{3}{5}+\frac{7}{10} \right)}-\frac{1}{2}=\frac{3}{5}+\textcolor{#2ab7ec}{\left(\frac{7}{10}-\frac{1}{2} \right)}=\frac{3}{5}+\frac{7}{10}-\frac{1}{2}}
Every difference can be written as a sum.
E.g.:
\latex{3-5=3+(-5)}
\latex{(a+b)+ c=\\=a+ (b+ c)=\\=a+ b+ c}
  1. The factors can be grouped arbitrarily.
\latex{\begin{rcases}\begin{alinged} \textcolor{#2ab7ec}{\left(\dfrac{1}{\underset{1}{\cancel{3}}}\times \dfrac{\overset{2}{\cancel{6}}}{5} \right)}\times \dfrac{5}{2}=\dfrac{\overset{1}{\cancel{2}}}{\underset{1}{\cancel{5}}}\times \frac{\overset{1}{\cancel{5}}}{\underset{1}{\cancel{2}}}=1 \\ \dfrac{1}{3}\times \textcolor{#2ab7ec}{\left(\dfrac{\overset{3}{\cancel{6}}}{\underset{1}{\cancel{5}} }\times \dfrac{\overset{1}{\cancel{5}}}{\underset{1}{\cancel{2}}} \right)}=\dfrac{1}{\underset{1}{\cancel{3}}}\times \overset{1}{\cancel{3}}=1 \\ \dfrac{1}{\underset{1}{\cancel{3}}}\times \dfrac{\overset{\overset{1}{\cancel{2}}}{\cancel{6}}}{\overset{1}{\cancel{5}}}\times \dfrac{\overset{1}{\cancel{5}}}{\underset{1}{\cancel{2}}}=1 \end{aligned}\end{rcases} \textcolor{#2ab7ec}{\left(\frac{1}{3}\times \dfrac{6}{5} \right)}\times \dfrac{5}{2}=\dfrac{1}{3}\times \textcolor{#2ab7ec}{\left(\dfrac{6}{5}\times \dfrac{5}{2} \right)}=\dfrac{1}{3}\times \dfrac{6}{5}\times \dfrac{5}{2}}
\latex{(a\times b)\times c=\\=a\times (b\times c)=\\=a\times b\times c}
  1. Multiplication of sums and differences can be done by multiplying the terms separately and then adding or subtracting them.
\latex{\begin{rcases}\begin{aligned} \dfrac{2}{5}\times \left(\dfrac{10}{3}+\dfrac{5}{6} \right)=\dfrac{2}{5}\times \left(\dfrac{20}{6}+\dfrac{5}{6} \right)=\dfrac{\overset{1}{\cancel{2}}}{\underset{1}{\cancel{5}}} \times \dfrac{\overset{5}{\cancel{25}}}{\underset{3}{\cancel{6}}}&=\dfrac{5}{3} \\ \dfrac{2}{5}\times \dfrac{10}{3}+\dfrac{2}{5}\times \dfrac{5}{6}=\dfrac{\overset{2}{\cancel{10}}}{3}\times \dfrac{2}{\underset{1}{\cancel{5}}}+\dfrac{\overset{1}{\cancel{5}}}{\underset{3}{\cancel{6}}}\times \dfrac{\overset{1}{\cancel{2}}}{\underset{1}{\cancel{5}}}=\dfrac{4}{3}+\dfrac{1}{3}&=\dfrac{5}{3} \end{aligned}\end{rcases}}
Note:
Sums and differences can be divided by a fraction by multiplying them with the reciprocal of the fraction.
\latex{(a+b)\times c=\\=a\times c+b\times c}
Exercises
{{exercise_number}}. Complete the operations
  1. \latex{ 8\times \left(\frac{1}{4}+\frac{1}{2}+\frac{5}{8} \right) };
  1. \latex{ \left(\frac{3}{32}+\frac{1}{80}+\frac{1}{50} \right)\times 16 };
  1. \latex{ \left(\frac{9}{10}-\frac{7}{30} \right)\times 10 };
  1. \latex{ \frac{2}{3} \times \left(\frac{3}{8}-\frac{3}{16} \right) };
  1. \latex{ \left(\frac{20}{9}-\frac{16}{15} \right)\times \frac{3}{4} };
  1. \latex{ \frac{27}{32}\times \left(\frac{5}{9}+\frac{1}{3} \right) }.
{{exercise_number}}. Are the results of the series of operations equal \latex{ (=) }, or not equal\latex{ (\neq ) }? Write the correct symbol in the square. Check your answers by calculating.
  1. \latex{ 5\times \frac{3}{11}+6\times \frac{3}{11} }   \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}   \latex{ \left(5+6\right)\times \frac{3}{11} }
  1. \latex{ 3\times \frac{3}{8}+\frac{1}{8}\times 3 }   \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}  \latex{ \left(3+\frac{1}{8} \right)\times 3 }
  1. \latex{ \frac{7}{8}\times 19-7\times \frac{7}{8} }      \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}   \latex{ \frac{7}{8} \times \left(19-7\right) }
  1. \latex{ \frac{7}{10}\times 5-\frac{7}{10}\times \frac{5}{14} }    \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}   \latex{ \left(5-\frac{5}{14} \right)\times \frac{7}{10} }
  1. \latex{ \frac{5}{4} \times 4+\frac{4}{5}\times \frac{5}{4} }       \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}   \latex{ \frac{4}{5}\times \left(\frac{5}{4}+4 \right) }
  1. \latex{ \left(2\frac{1}{2}-\frac{16}{5} \right)\times \frac{5}{11} }   \latex{\fcolorbox{black}{eaf1fa}{\phantom{--}}}   \latex{ 2\frac{1}{2}\times \frac{5}{11}-\frac{16}{5}\times \frac{5}{11} }
{{exercise_number}}. Arrange the results of the following operations in ascending order.
  1. \latex{\frac{3}{4}-\left(\frac{7}{6}-\frac{1}{2} \right) }
  1. \latex{\frac{3}{5}\div \left(\frac{1}{2}-\frac{1}{3} \right) }
  1. \latex{ \left(\frac{2}{3}-\frac{2}{5} \right)\times \frac{1}{2} }
  1. \latex{ \frac{7}{2}\times \left(\frac{6}{5}\div \frac{1}{2} \right)}
  1. \latex{\frac{1}{3}+\frac{1}{5}-\frac{1}{30} }
  1. \latex{\frac{8}{5}\div \frac{1}{2}\times \frac{1}{4} }
{{exercise_number}}. Rewrite the operations as the multiplication or division of a sum or a difference. Check your answer by calculating. Which is the simpler procedure in each case?
For example: \latex{\frac{2}{3}\times \frac{4}{5}+\frac{2}{3}\times \frac{1}{5}=\frac{2}{3}\times \left(\frac{4}{5}+\frac{1}{5} \right)}
  1. \latex{\frac{8}{9}\times \frac{3}{8}+\frac{8}{9}\times \frac{2}{8}}
  1. \latex{\frac{5}{2}\times \frac{2}{5}-\frac{5}{2}\times \frac{1}{10}}
  1. \latex{5\frac{7}{20}\times 20}
  1. \latex{\frac{14}{3}\div 2-\frac{8}{5}\div 2}
  1. \latex{\frac{7}{2}\div 11+\frac{6}{5}\div 11}
  1. \latex{\frac{1}{2}\div 1\frac{1}{2}+\frac{7}{8}\div 1\frac{1}{2}}
{{exercise_number}}. At a bakery, the flour needed to make \latex{ 22 } loaves of bread has been taken out of a \latex{ 25\;kg } bag.
How much flour is left in the bag if \latex{\frac{3}{4}\;kg} of flour is needed to bake one loaf of bread?
{{exercise_number}}. Sixth-grade students wore ribbons as part of their Halloween costumes. The girls wore an \latex{ 80 \;cm } long pink ribbon, while the boys had a half-\latex{ metre }-long blue ribbon. How many boys and girls are in the sixth grade if \latex{ 15.5\; metres } of ribbon were used in total?
{{exercise_number}}. Edith is knitting a scarf. She completes \latex{ 1\frac{3}{5}\ m} every \latex{ hour. } How many \latex{ metres } does the length of the scarf increase in \latex{ 6 } \latex{ days } if she knits for half an \latex{ hour } each \latex{ day? } Which of the following operations expresses the solution of the exercise?
  1. \latex{ \left(1\frac{3}{5}\times \frac{1}{2} \right)\times 6 }
  1. \latex{ 1\frac{3}{5}\div \frac{1}{2} \times 6 }
  1. \latex{ \left(1\frac{3}{5}\times 6 \right)\div 2 }
  1. \latex{ 1\frac{3}{5}\div \left(2\times 6\right) }
  1. \latex{ 1\frac{3}{5}\times \frac{1}{2}\times 6 }
  1. \latex{ \left(1\frac{3}{5}\div 2 \right)\times 6 }
{{exercise_number}}. Write a word problem whose solution is
  1. \latex{8-\left(3+2\frac{1}{2} \right);}
  1. \latex{8-\left(3+2\frac{1}{2} \right);}
  1. \latex{\left(\frac{4}{5}+\frac{1}{2} \right)\times 2\frac{1}{2};}
  1. \latex{\frac{4}{5}\times 2\frac{1}{2}+\frac{1}{2}} .
Quiz
Organise the cards to make the equality true.
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ + }
\latex{ \div }
\latex{ = }
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